Properties

Label 4032.2.b.d.3583.2
Level 4032
Weight 2
Character 4032.3583
Analytic conductor 32.196
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.2
Root \(2.44949i\)
Character \(\chi\) = 4032.3583
Dual form 4032.2.b.d.3583.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.44949i q^{5} +(-1.00000 + 2.44949i) q^{7} +O(q^{10})\) \(q+2.44949i q^{5} +(-1.00000 + 2.44949i) q^{7} -2.44949i q^{11} -4.89898i q^{13} -2.44949i q^{17} +2.00000 q^{19} +7.34847i q^{23} -1.00000 q^{25} +6.00000 q^{29} +8.00000 q^{31} +(-6.00000 - 2.44949i) q^{35} -4.00000 q^{37} +7.34847i q^{41} -4.89898i q^{43} +12.0000 q^{47} +(-5.00000 - 4.89898i) q^{49} -6.00000 q^{53} +6.00000 q^{55} +12.0000 q^{59} +12.0000 q^{65} +12.2474i q^{71} -14.6969i q^{73} +(6.00000 + 2.44949i) q^{77} +9.79796i q^{79} +6.00000 q^{85} +12.2474i q^{89} +(12.0000 + 4.89898i) q^{91} +4.89898i q^{95} +4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 4q^{19} - 2q^{25} + 12q^{29} + 16q^{31} - 12q^{35} - 8q^{37} + 24q^{47} - 10q^{49} - 12q^{53} + 12q^{55} + 24q^{59} + 24q^{65} + 12q^{77} + 12q^{85} + 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.44949i 1.09545i 0.836660 + 0.547723i \(0.184505\pi\)
−0.836660 + 0.547723i \(0.815495\pi\)
\(6\) 0 0
\(7\) −1.00000 + 2.44949i −0.377964 + 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949i 0.738549i −0.929320 0.369274i \(-0.879606\pi\)
0.929320 0.369274i \(-0.120394\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i −0.733799 0.679366i \(-0.762255\pi\)
0.733799 0.679366i \(-0.237745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.44949i 0.594089i −0.954864 0.297044i \(-0.903999\pi\)
0.954864 0.297044i \(-0.0960009\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.34847i 1.53226i 0.642685 + 0.766131i \(0.277821\pi\)
−0.642685 + 0.766131i \(0.722179\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 2.44949i −1.01419 0.414039i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.34847i 1.14764i 0.818982 + 0.573819i \(0.194539\pi\)
−0.818982 + 0.573819i \(0.805461\pi\)
\(42\) 0 0
\(43\) 4.89898i 0.747087i −0.927613 0.373544i \(-0.878143\pi\)
0.927613 0.373544i \(-0.121857\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −5.00000 4.89898i −0.714286 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.2474i 1.45350i 0.686900 + 0.726752i \(0.258971\pi\)
−0.686900 + 0.726752i \(0.741029\pi\)
\(72\) 0 0
\(73\) 14.6969i 1.72015i −0.510171 0.860073i \(-0.670418\pi\)
0.510171 0.860073i \(-0.329582\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 + 2.44949i 0.683763 + 0.279145i
\(78\) 0 0
\(79\) 9.79796i 1.10236i 0.834388 + 0.551178i \(0.185822\pi\)
−0.834388 + 0.551178i \(0.814178\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2474i 1.29823i 0.760692 + 0.649113i \(0.224860\pi\)
−0.760692 + 0.649113i \(0.775140\pi\)
\(90\) 0 0
\(91\) 12.0000 + 4.89898i 1.25794 + 0.513553i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.89898i 0.502625i
\(96\) 0 0
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.44949i 0.243733i −0.992546 0.121867i \(-0.961112\pi\)
0.992546 0.121867i \(-0.0388880\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.2474i 1.18401i 0.805936 + 0.592003i \(0.201663\pi\)
−0.805936 + 0.592003i \(0.798337\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 + 2.44949i 0.550019 + 0.224544i
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796i 0.876356i
\(126\) 0 0
\(127\) 9.79796i 0.869428i −0.900568 0.434714i \(-0.856849\pi\)
0.900568 0.434714i \(-0.143151\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −2.00000 + 4.89898i −0.173422 + 0.424795i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 14.6969i 1.22051i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 14.6969i 1.19602i 0.801489 + 0.598010i \(0.204042\pi\)
−0.801489 + 0.598010i \(0.795958\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 19.5959i 1.57398i
\(156\) 0 0
\(157\) 19.5959i 1.56392i −0.623326 0.781962i \(-0.714219\pi\)
0.623326 0.781962i \(-0.285781\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −18.0000 7.34847i −1.41860 0.579141i
\(162\) 0 0
\(163\) 19.5959i 1.53487i 0.641126 + 0.767435i \(0.278467\pi\)
−0.641126 + 0.767435i \(0.721533\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.34847i 0.558694i 0.960190 + 0.279347i \(0.0901179\pi\)
−0.960190 + 0.279347i \(0.909882\pi\)
\(174\) 0 0
\(175\) 1.00000 2.44949i 0.0755929 0.185164i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.1464i 1.28158i 0.767714 + 0.640792i \(0.221394\pi\)
−0.767714 + 0.640792i \(0.778606\pi\)
\(180\) 0 0
\(181\) 4.89898i 0.364138i 0.983286 + 0.182069i \(0.0582795\pi\)
−0.983286 + 0.182069i \(0.941721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.79796i 0.720360i
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.44949i 0.177239i −0.996066 0.0886194i \(-0.971755\pi\)
0.996066 0.0886194i \(-0.0282455\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 + 14.6969i −0.421117 + 1.03152i
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.89898i 0.338869i
\(210\) 0 0
\(211\) 4.89898i 0.337260i 0.985679 + 0.168630i \(0.0539342\pi\)
−0.985679 + 0.168630i \(0.946066\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) −8.00000 + 19.5959i −0.543075 + 1.33026i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 24.4949i 1.61867i 0.587348 + 0.809334i \(0.300172\pi\)
−0.587348 + 0.809334i \(0.699828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 30.0000 1.96537 0.982683 0.185296i \(-0.0593245\pi\)
0.982683 + 0.185296i \(0.0593245\pi\)
\(234\) 0 0
\(235\) 29.3939i 1.91745i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.2474i 0.792222i −0.918203 0.396111i \(-0.870360\pi\)
0.918203 0.396111i \(-0.129640\pi\)
\(240\) 0 0
\(241\) 4.89898i 0.315571i 0.987473 + 0.157786i \(0.0504355\pi\)
−0.987473 + 0.157786i \(0.949565\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.0000 12.2474i 0.766652 0.782461i
\(246\) 0 0
\(247\) 9.79796i 0.623429i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.0454i 1.37515i −0.726111 0.687577i \(-0.758674\pi\)
0.726111 0.687577i \(-0.241326\pi\)
\(258\) 0 0
\(259\) 4.00000 9.79796i 0.248548 0.608816i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.44949i 0.151042i 0.997144 + 0.0755210i \(0.0240620\pi\)
−0.997144 + 0.0755210i \(0.975938\pi\)
\(264\) 0 0
\(265\) 14.6969i 0.902826i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.0454i 1.34413i −0.740491 0.672066i \(-0.765407\pi\)
0.740491 0.672066i \(-0.234593\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.44949i 0.147710i
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.0000 7.34847i −1.06251 0.433766i
\(288\) 0 0
\(289\) 11.0000 0.647059
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.0454i 1.28791i −0.765065 0.643953i \(-0.777293\pi\)
0.765065 0.643953i \(-0.222707\pi\)
\(294\) 0 0
\(295\) 29.3939i 1.71138i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 12.0000 + 4.89898i 0.691669 + 0.282372i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 9.79796i 0.553813i −0.960897 0.276907i \(-0.910691\pi\)
0.960897 0.276907i \(-0.0893093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 14.6969i 0.822871i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.89898i 0.272587i
\(324\) 0 0
\(325\) 4.89898i 0.271746i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 + 29.3939i −0.661581 + 1.62054i
\(330\) 0 0
\(331\) 24.4949i 1.34636i −0.739478 0.673181i \(-0.764928\pi\)
0.739478 0.673181i \(-0.235072\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 19.5959i 1.06118i
\(342\) 0 0
\(343\) 17.0000 7.34847i 0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.1464i 0.920468i 0.887798 + 0.460234i \(0.152235\pi\)
−0.887798 + 0.460234i \(0.847765\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i 0.851427 + 0.524473i \(0.175738\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.44949i 0.130373i 0.997873 + 0.0651866i \(0.0207643\pi\)
−0.997873 + 0.0651866i \(0.979236\pi\)
\(354\) 0 0
\(355\) −30.0000 −1.59223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.2474i 0.646396i 0.946331 + 0.323198i \(0.104758\pi\)
−0.946331 + 0.323198i \(0.895242\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 14.6969i 0.311504 0.763027i
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 29.3939i 1.51386i
\(378\) 0 0
\(379\) 34.2929i 1.76151i 0.473576 + 0.880753i \(0.342963\pi\)
−0.473576 + 0.880753i \(0.657037\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −6.00000 + 14.6969i −0.305788 + 0.749025i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) 9.79796i 0.491745i −0.969302 0.245873i \(-0.920925\pi\)
0.969302 0.245873i \(-0.0790745\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 39.1918i 1.95228i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) 4.89898i 0.242239i 0.992638 + 0.121119i \(0.0386484\pi\)
−0.992638 + 0.121119i \(0.961352\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.0000 + 29.3939i −0.590481 + 1.44638i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.44949i 0.118818i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.1464i 0.825914i 0.910750 + 0.412957i \(0.135504\pi\)
−0.910750 + 0.412957i \(0.864496\pi\)
\(432\) 0 0
\(433\) 9.79796i 0.470860i −0.971891 0.235430i \(-0.924350\pi\)
0.971891 0.235430i \(-0.0756498\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.6969i 0.703050i
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.44949i 0.116379i 0.998306 + 0.0581894i \(0.0185327\pi\)
−0.998306 + 0.0581894i \(0.981467\pi\)
\(444\) 0 0
\(445\) −30.0000 −1.42214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 + 29.3939i −0.562569 + 1.37801i
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.44949i 0.114084i 0.998372 + 0.0570421i \(0.0181669\pi\)
−0.998372 + 0.0570421i \(0.981833\pi\)
\(462\) 0 0
\(463\) 9.79796i 0.455350i −0.973737 0.227675i \(-0.926888\pi\)
0.973737 0.227675i \(-0.0731123\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 19.5959i 0.893497i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 4.89898i 0.221994i −0.993821 0.110997i \(-0.964596\pi\)
0.993821 0.110997i \(-0.0354044\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.7423i 1.65816i −0.559131 0.829079i \(-0.688865\pi\)
0.559131 0.829079i \(-0.311135\pi\)
\(492\) 0 0
\(493\) 14.6969i 0.661917i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −30.0000 12.2474i −1.34568 0.549373i
\(498\) 0 0
\(499\) 14.6969i 0.657925i −0.944343 0.328963i \(-0.893301\pi\)
0.944343 0.328963i \(-0.106699\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.1464i 0.760002i 0.924986 + 0.380001i \(0.124076\pi\)
−0.924986 + 0.380001i \(0.875924\pi\)
\(510\) 0 0
\(511\) 36.0000 + 14.6969i 1.59255 + 0.650154i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 39.1918i 1.72700i
\(516\) 0 0
\(517\) 29.3939i 1.29274i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.44949i 0.107314i 0.998559 + 0.0536570i \(0.0170878\pi\)
−0.998559 + 0.0536570i \(0.982912\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.5959i 0.853612i
\(528\) 0 0
\(529\) −31.0000 −1.34783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) −30.0000 −1.29701
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 + 12.2474i −0.516877 + 0.527535i
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.89898i 0.209849i
\(546\) 0 0
\(547\) 29.3939i 1.25679i −0.777894 0.628396i \(-0.783712\pi\)
0.777894 0.628396i \(-0.216288\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −24.0000 9.79796i −1.02058 0.416652i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 14.6969i 0.618305i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 19.5959i 0.820064i −0.912071 0.410032i \(-0.865518\pi\)
0.912071 0.410032i \(-0.134482\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.34847i 0.306452i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14.6969i 0.608685i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 46.5403i 1.91118i 0.294697 + 0.955591i \(0.404781\pi\)
−0.294697 + 0.955591i \(0.595219\pi\)
\(594\) 0 0
\(595\) −6.00000 + 14.6969i −0.245976 + 0.602516i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.44949i 0.100083i 0.998747 + 0.0500417i \(0.0159354\pi\)
−0.998747 + 0.0500417i \(0.984065\pi\)
\(600\) 0 0
\(601\) 19.5959i 0.799334i 0.916660 + 0.399667i \(0.130874\pi\)
−0.916660 + 0.399667i \(0.869126\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.2474i 0.497930i
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 58.7878i 2.37830i
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.0000 12.2474i −1.20192 0.490684i
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.79796i 0.390670i
\(630\) 0 0
\(631\) 19.5959i 0.780101i 0.920793 + 0.390051i \(0.127542\pi\)
−0.920793 + 0.390051i \(0.872458\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) −24.0000 + 24.4949i −0.950915 + 0.970523i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 29.3939i 1.15381i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.34847i 0.286256i −0.989704 0.143128i \(-0.954284\pi\)
0.989704 0.143128i \(-0.0457160\pi\)
\(660\) 0 0
\(661\) 29.3939i 1.14329i 0.820501 + 0.571645i \(0.193694\pi\)
−0.820501 + 0.571645i \(0.806306\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 4.89898i −0.465340 0.189974i
\(666\) 0 0
\(667\) 44.0908i 1.70720i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.2474i 0.470708i 0.971910 + 0.235354i \(0.0756249\pi\)
−0.971910 + 0.235354i \(0.924375\pi\)
\(678\) 0 0
\(679\) −12.0000 4.89898i −0.460518 0.188006i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.34847i 0.281181i −0.990068 0.140591i \(-0.955100\pi\)
0.990068 0.140591i \(-0.0449002\pi\)
\(684\) 0 0
\(685\) 14.6969i 0.561541i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 29.3939i 1.11982i
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.79796i 0.371658i
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 + 2.44949i 0.225653 + 0.0921225i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 58.7878i 2.20162i
\(714\) 0 0
\(715\) 29.3939i 1.09927i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 16.0000 39.1918i 0.595871 1.45958i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 4.89898i 0.180948i −0.995899 0.0904740i \(-0.971162\pi\)
0.995899 0.0904740i \(-0.0288382\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 9.79796i 0.360424i 0.983628 + 0.180212i \(0.0576783\pi\)
−0.983628 + 0.180212i \(0.942322\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.2474i 0.449315i −0.974438 0.224658i \(-0.927874\pi\)
0.974438 0.224658i \(-0.0721264\pi\)
\(744\) 0 0
\(745\) 44.0908i 1.61536i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −30.0000 12.2474i −1.09618 0.447512i
\(750\) 0 0
\(751\) 34.2929i 1.25136i −0.780078 0.625682i \(-0.784821\pi\)
0.780078 0.625682i \(-0.215179\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.0000 −1.31017
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.6413i 1.50950i −0.656014 0.754748i \(-0.727759\pi\)
0.656014 0.754748i \(-0.272241\pi\)
\(762\) 0 0
\(763\) 2.00000 4.89898i 0.0724049 0.177355i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 58.7878i 2.12270i
\(768\) 0 0
\(769\) 9.79796i 0.353323i −0.984272 0.176662i \(-0.943470\pi\)
0.984272 0.176662i \(-0.0565299\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.0454i 0.792918i 0.918052 + 0.396459i \(0.129761\pi\)
−0.918052 + 0.396459i \(0.870239\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.6969i 0.526572i
\(780\) 0 0
\(781\) 30.0000 1.07348
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 48.0000 1.71319
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 14.6969i 0.213335 0.522563i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.7423i 1.30148i −0.759300 0.650740i \(-0.774459\pi\)
0.759300 0.650740i \(-0.225541\pi\)
\(798\) 0 0
\(799\) 29.3939i 1.03988i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −36.0000 −1.27041
\(804\) 0 0
\(805\) 18.0000 44.0908i 0.634417 1.55400i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) 9.79796i 0.342787i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)